US 8165854 B1 Abstract Methods, systems, and related computer program products for photolithographic process simulation are disclosed. In one preferred embodiment, a resist processing system is simulated according to a Wiener nonlinear model thereof in which a plurality of precomputed optical intensity distributions corresponding to a respective plurality of distinct elevations in an optically exposed resist film are received, each optical intensity distribution is convolved with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results, and at least two of the convolution results are multiplied to produce at least one cross-product. A weighted summation of the plurality of convolution results and the at least one cross-product is computed using a respective plurality of predetermined Wiener coefficients to generate a Wiener output, and a resist processing system simulation result is generated based at least in part on the Wiener output.
Claims(46) 1. A method for simulating a resist processing system according to a Wiener nonlinear model thereof, comprising:
receiving a plurality of precomputed optical intensity distributions corresponding to a respective plurality of distinct elevations in an optically exposed resist film;
convolving each of said optical intensity distributions with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results;
cross-multiplying at least two of said convolution results to produce at least one cross-product;
computing a first weighted summation of said plurality of convolution results and said at least one cross-product using a respective first plurality of predetermined Wiener coefficients to generate a first Wiener output; and
generating a resist processing system simulation result based at least in part on said first Wiener output.
2. The method of
3. The method of
4. The method of
5. The method of
6. The method of
7. The method of
8. The method of
9. A method for calibrating a plurality of Wiener coefficients for use in a Wiener nonlinear model-based computer simulation of a resist processing system, comprising:
receiving first information representative of a reference developed resist structure associated with a test mask design;
receiving second information representative of a plurality of precomputed optical intensity distributions corresponding to respective distinct elevations of an optical exposure pattern associated with said test mask design;
convolving each of said optical intensity distributions with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results;
cross-multiplying at least two of said convolution results to produce at least one cross-product;
initializing the plurality of Wiener coefficients;
computing with a processing system a weighted summation of said plurality of convolution results and said at least one cross-product using said plurality of Wiener coefficients to generate a current Wiener output;
processing said current Wiener output to generate third information representative of a current virtual developed resist structure;
modifying said plurality of Wiener coefficients based on said first information and said third information; and
repeating said computing the weighted summation, said processing the current Wiener output, and said modifying the plurality of Wiener coefficients until a small error condition is reached between said third information and said first information to produce the calibrated Wiener coefficients.
10. The method of
exposing a physical resist film disposed on a wafer using a physical optical exposure system and a physical test mask based on said test mask design; and
physically processing the exposed physical resist film using said physical version of the resist processing system to generate said reference developed resist structure.
11. The method of
12. The method of
13. The method of
14. The method of
receiving said second information representative of the plurality of precomputed optical intensity distributions; and
processing said plurality of precomputed optical intensity distributions using an antecedent simulation of the resist processing system to generate an antecedent simulator output; and
extracting said first information from said antecedent simulator output.
15. The method of
differential-equation-solving simulators, finite difference simulators, finite element simulators, and closed-form model simulators.
16. The method of
_{W}>2, and wherein said antecedent simulation of the resist processing system is based on a Wiener nonlinear model incorporating an antecedent convolution result cross-product computation having a highest order ORD_{ANTECEDENT }that is less than ORD_{W}.17. The method of
18. The method of
19. A method for simulating an etch system according to a Wiener nonlinear model thereof, comprising:
receiving first information representative of a plurality of precomputed developed resist distributions corresponding to a respective plurality of distinct elevations in a developed resist structure disposed on a wafer;
convolving each of said developed resist distributions with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results;
cross-multiplying at least two of said convolution results to produce at least one cross-product;
computing a first weighted summation of said plurality of convolution results and said at least one cross-product using a respective first plurality of predetermined Wiener coefficients to generate a first Wiener output; and
generating an etch system simulation result based at least in part on said first Wiener output.
20. The method of
21. The method of
22. The method of
23. The method of
24. The method of
25. A method for calibrating a plurality of Wiener coefficients for use in a Wiener nonlinear model-based computer simulation of an etch system, comprising:
receiving first information representative of a reference etched wafer structure associated with a precomputed test developed resist structure;
receiving second information representative of a plurality of developed resist distributions corresponding to a respective plurality of distinct elevations in the precomputed test developed resist structure;
convolving each of said developed resist distributions with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results;
cross-multiplying at least two of said convolution results to produce at least one cross-product;
initializing the plurality of Wiener coefficients;
computing with a processing system a weighted summation of said plurality of convolution results and said at least one cross-product using said plurality of Wiener coefficients to generate a current Wiener output;
processing said current Wiener output to generate third information representative of a current virtual etched wafer structure;
modifying said plurality of Wiener coefficients based on said first information and said third information; and
repeating said computing the weighted summation, said processing the current Wiener output, and said modifying the plurality of Wiener coefficients until a small error condition is reached between said third information and said first information to produce the calibrated Wiener coefficients.
26. The method of
receiving said second information representative of the plurality of developed resist distributions; and
processing said plurality of developed resist distributions using an antecedent simulation of the etch system to generate an antecedent simulator output; and
extracting said first information from said antecedent simulator output.
27. The method of
28. The method of
_{W}>2, and wherein said antecedent simulation of the etch system is based on a Wiener nonlinear model incorporating an antecedent convolution result cross-product computation having a highest order ORD_{ANTECEDENT }that is less than ORD_{W}.29. The method of
30. A method for simulating a resist processing system, comprising:
receiving a plurality of precomputed optical intensity distributions corresponding to a respective plurality of distinct elevations in an optically exposed resist film;
filtering of each of said optical intensity distributions with each of a plurality of predetermined filters to generate a plurality of filtering results, said predetermined filters corresponding to a closed-form mathematical model of the resist processing system;
computing at least one nonlinear function of at least two of said filtering results to produce at least one nonlinear result;
computing a weighted combination of said plurality of filtering results and said at least one nonlinear result using a respective plurality of predetermined weighting coefficients associated with said closed-form mathematical model to generate a model output; and
generating a resist processing system simulation result based at least in part on said model output.
31. A method for calibrating a plurality of weighting coefficients for use in a closed-form mathematical model-based computer simulation of a resist processing system, comprising:
receiving first information representative of a reference developed resist structure associated with a test mask design;
receiving second information representative of a plurality of precomputed optical intensity distributions corresponding to respective distinct elevations of an optical exposure pattern associated with said test mask design;
filtering of each of said optical intensity distributions with each of a plurality of predetermined filters associated with said closed-form mathematical model to generate a plurality of filtering results;
computing with a processing system at least one nonlinear function of at least two of said filtering results to produce at least one nonlinear result;
initializing the plurality of weighting coefficients;
computing a weighted combination of said plurality of filtering results and said at least one nonlinear result using said plurality of weighting coefficients to generate a current model output;
processing said current model output to generate third information representative of a current virtual developed resist structure;
modifying said plurality of weighting coefficients based on said first information and said third information; and
repeating said computing the weighted summation, said processing the current model output, and said modifying the plurality of weighting coefficients until a small error condition is reached between said third information and said first information to produce the calibrated weighting coefficients.
32. A method for simulating an etch system, comprising:
receiving first information representative of a plurality of precomputed developed resist distributions corresponding to a respective plurality of distinct elevations in a developed resist structure disposed on a wafer;
filtering of each of said developed resist distributions with each of a plurality of predetermined filters to generate a plurality of filtering results, said predetermined filters corresponding to a closed-form mathematical model of the etch system;
computing with a processing system at least one nonlinear function of at least two of said filtering results to produce at least one nonlinear result;
computing a first weighted combination of said plurality of filtering results and said at least one nonlinear result using a respective plurality of predetermined weighting coefficients to generate a model output; and
generating an etch system simulation result based at least in part on said model output.
33. A method for calibrating a plurality of weighting coefficients for use in a closed-form mathematical model-based computer simulation of an etch system, comprising:
receiving first information representative of a reference etched wafer structure associated with a precomputed test developed resist structure;
receiving second information representative of a plurality of developed resist distributions corresponding to a respective plurality of distinct elevations in the precomputed test developed resist structure;
filtering each of said developed resist distributions with each of a plurality of predetermined filters associated with said closed-form mathematical model to generate a plurality of filtering results;
computing with a processing system at least one nonlinear function of at least two of said filtering results to produce at least one nonlinear result;
initializing the plurality of weighting coefficients;
computing a weighted summation of said plurality of filtering results and said at least one nonlinear result using said plurality of weighting coefficients to generate a current model output;
processing said current model output to generate third information representative of a current virtual etched wafer structure;
modifying said plurality of weighting coefficients based on said first information and said third information; and
repeating said computing the weighted summation, said processing the current model output, and said modifying the plurality of weighting coefficients until a small error condition is reached between said third information and said first information to produce the calibrated weighting coefficients.
34. A method for simulating an optical exposure system in which optical radiation incident upon a photomask is modulated thereby and projected toward a target to generate a target intensity pattern, the optical radiation incident upon the photomask comprising a plurality of spatial frequency components, the method comprising:
receiving a plurality of mask layer distribution functions representative of a respective plurality of layers of the photomask;
receiving first information representative of a first of the plurality of spatial frequency components of the incident optical radiation;
convolving each of said mask layer distribution functions with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results;
cross-multiplying at least two of said convolution results to produce at least one cross-product;
computing a first weighted summation of said plurality of convolution results and said at least one cross-product using a respective first plurality of predetermined Wiener coefficients to generate a first Wiener output representative of a first modulated radiation result associated with said first spatial frequency component; and
generating a target intensity pattern based at least in part on said first modulated radiation result.
35. The method of
receiving second information representative of a second of the plurality of spatial frequency components of the incident optical radiation; and
computing a second weighted summation of said plurality of convolution results and said at least one cross-product using a respective second plurality of predetermined Wiener coefficients to generate a second Wiener output representative of a second modulated radiation result associated with said second spatial frequency component;
wherein said target intensity pattern is further based at least in part on said second modulated radiation result.
36. The method of
processing said first modulated radiation result according to a projection simulation algorithm to generate a first target partial intensity distribution corresponding thereto;
processing said second modulated radiation result according to said projection simulation algorithm to generate a second target partial intensity distribution corresponding thereto; and
computing said target intensity pattern as a sum of said first and second target partial intensity distributions.
37. A method for calibrating a plurality of Wiener coefficients for use in a Wiener nonlinear model-based computer simulation of photomask diffraction of incident optical radiation at a selected spatial frequency, comprising:
receiving first information representative of a reference modulated radiation result associated with a test photomask and said selected spatial frequency;
receiving a plurality of mask layer distribution functions representative of a respective plurality of layers of the test photomask;
convolving each of said mask layer distribution functions with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results;
cross-multiplying at least two of said convolution results to produce at least one cross-product;
initializing the plurality of Wiener coefficients;
computing with a processing system a weighted summation of said plurality of convolution results and said at least one cross-product using said plurality of Wiener coefficients to generate a current Wiener output representative of a current virtual modulated radiation result;
modifying said plurality of Wiener coefficients based on said reference modulated radiation result and said current virtual modulated radiation result; and
repeating said computing the weighted summation and said modifying the plurality of Wiener coefficients until a small error condition is reached between said current modulated radiation result and said reference modulated radiation result to produce the calibrated Wiener coefficients.
38. The method of
receiving second information representative of said plurality of layers of the test photomask;
processing said second information using an antecedent simulation of the photomask diffraction of the incident optical radiation at said selected spatial frequency to generate said first information.
39. The method of
40. The method of
_{W}>2, and wherein said antecedent simulation is based on a Wiener nonlinear model incorporating an antecedent convolution result cross-product computation having a highest order ORD_{ANTECEDENT }that is less than ORD_{W}.41. A method for simulating a resist processing system according to a Wiener nonlinear model thereof, comprising:
receiving a plurality of precomputed optical intensity distributions corresponding to a respective plurality of distinct elevations in an optically exposed resist film;
convolving each of said optical intensity distributions with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results;
cross-multiplying at least two of said convolution results to produce at least one cross-product;
receiving a plurality of process variation factors associated with the resist processing system;
computing each of a plurality of Wiener coefficients as a respective predetermined polynomial function of said process variation factors characterized by a respective distinct set of predetermined polynomial coefficients;
computing a weighted summation of said plurality of convolution results and said at least one cross-product using said computed plurality of Wiener coefficients to generate a Wiener output; and
generating a resist processing system simulation result based at least in part on said Wiener output.
42. The method of
43. The method of
44. A method for calibrating a plurality of sets of polynomial coefficients for use in a Wiener nonlinear model-based computer simulation of a resist processing system, comprising:
receiving first information representative of a plurality of reference developed resist structures, said reference developed resist structures being associated with a common test mask design but each said reference developed resist structure being associated with a respective one of a known plurality of distinctly valued process variation factor sets associated with the resist processing system;
receiving second information representative of a plurality of precomputed optical intensity distributions corresponding to respective distinct elevations of an optical exposure pattern associated with said common test mask design;
cross-multiplying at least two of said convolution results to produce at least one cross-product;
initializing the plurality of sets of polynomial coefficients;
computing with a processing system a plurality of current Wiener outputs associated with respective ones of said distinctly valued process variation factor sets, wherein, for each distinctly valued process variation factor set, said computing the current Wiener output comprises:
computing each of a plurality of Wiener coefficients as a respective predetermined polynomial function of said process variation factors, each said predetermined polynomial function using a respective one of said sets of polynomial coefficients; and
computing a weighted summation of said plurality of convolution results and said at least one cross-product using said computed plurality of Wiener coefficients to generate the current Wiener output;
processing said plurality of current Wiener outputs to generate third information representative of a respective plurality of current virtual developed resist structures;
modifying said plurality of sets of polynomial coefficients based on said first information and said third information; and
repeating said computing the plurality of current Wiener outputs, said processing said plurality of current Wiener outputs, and said modifying the plurality of sets of polynomial coefficients until a small error condition is reached between said third information and said first information to produce the calibrated sets of polynomial coefficients.
45. The method of
46. The method of
receiving said second information representative of the plurality of precomputed optical intensity distributions; and
for each one of said distinctly valued process variation factor sets, processing said second information using an antecedent simulation of the resist processing system to generate fourth information representative of a respective one of the plurality of reference developed resist structures, said first information being based at least in part on said fourth information.
Description This patent application claims the benefit of U.S. Provisional Ser. No. 60/948,467 filed Jul. 8, 2007, which is incorporated by reference herein. This patent application is a continuation-in-part of U.S. Ser. No. 11/708,444 filed Feb. 20, 2007, now abandoned, which is a continuation-in-part of U.S. Ser. No. 11/331,223, filed Jan. 11, 2006, now abandoned, each of which is incorporated by reference herein. This patent specification relates to computer simulation of photolithographic processes as may be employed, for example, in sub-wavelength integrated circuit fabrication environments. Computer simulation has become an indispensable tool in a wide variety of technical endeavors ranging from the design of aircraft, automobiles, and communications networks to the analysis of biological systems, socioeconomic trends, and plate tectonics. In the field of integrated circuit fabrication, computer simulation has become increasingly important as circuit line widths continue to shrink well below the wavelength of light. In particular, the optical projection of circuit patterns onto semiconductor wafers, during a process known as photolithography, becomes increasingly complicated to predict as pattern sizes shrink well below the wavelength of the light that is used to project the pattern. Historically, when circuit line widths were larger than the light wavelength, a desired circuit pattern was directly written to an optical mask, the mask was illuminated and projected toward the wafer, the circuit pattern was faithfully recorded in a layer of photoresist on the wafer, and, after chemical processing, the circuit pattern was faithfully realized in physical form on the wafer. However, for sub-wavelength circuit line widths, it becomes necessary to “correct” or pre-compensate the mask pattern in order for the desired circuit pattern to be properly recorded into the photoresist layer and/or for the proper physical form of the circuit pattern to appear on the wafer after chemical processing. Unfortunately, the required “corrections” or pre-compensations are themselves difficult to refine and, although there are some basic pre-compensation rules that a human designer can start with, the pre-compensation process is usually iterative (i.e., trial and error) and pattern-specific to the particular desired circuit pattern. Because human refinement and physical trial-and-error quickly become prohibitively expensive, optical proximity correction (OPC) software tools have been developed for the automation (optionally with a certain amount of human analysis and interaction along the way) of pre-compensating a desired circuit pattern before it is physically written onto a mask. Starting with a desired circuit pattern, an initial mask design is generated using a set of pre-compensation rules. For the initial mask design, together with a set of process conditions for an actual photolithographic processing system (e.g., a set of illumination/projection conditions/assumptions for a “stepper,” a set of conditions/assumptions for the subsequent resist processing track, a set of conditions/assumptions for the subsequent etching process), a simulation is performed that generates simulated images of (i) the developed resist structure on the wafer that would appear after the resist processing track, and/or (ii) the etched wafer structure after the etching process. The simulated images are compared to the desired circuit pattern, and deviations from the desired circuit pattern are determined. The mask design is then modified based on the deviations, and the simulation is repeated for the modified mask design. Deviations from the desired circuit pattern are again determined, and so on, the mask design being iteratively modified until the simulated images agree with the desired circuit pattern to within an acceptable tolerance. The accuracy of the simulated images are, of course, crucial in obtaining OPC-generated mask designs that lead to acceptable results in the actual production stepper machines, resist processing tracks, and etching systems of the actual integrated circuit fabrication environments. Notably, a typical thickness of the resist film While certain prior art approaches such as those based on differential-equation-solving techniques, finite difference techniques, or finite element techniques can provide accurate results, even to the extent of representing a kind of gold standard in their accuracy, they are generally quite slow. Approaches based on closed-form techniques avoid such timewise-recursive computation approaches and can therefore be much faster. However, many prior art closed-form techniques are believed to suffer from one or more shortcomings that are addressed by one or more of the preferred embodiments described hereinbelow. For example, many prior art closed-form techniques either ignore the third dimension altogether or are based on mathematical models do not adequately accommodate for the real-world, high aspect ratio, three-dimensional character of the interactions taking place. Other issues arise as would be apparent to one skilled in the art upon reading the present disclosure. By way of example, many known optical exposure system simulators operate on a simplifying assumption that the mask is a purely two-dimensional planar structure, whereas actual masks have some degree of thickness that affects the way the incident optical radiation is modulated. By way of further example, many of the above prior art techniques are unable to efficiently accommodate process variations in the photolithographic systems being modeled, and thus for each combination of values for physical variations in the photolithographic process (such as bake temperature, resist development time, etchant pH, stepper defocus parameter, etc.), such techniques require the entire (or almost entire) simulation process to be repeated essentially from the beginning. Provided in accordance with one or more of the preferred embodiments are methods, systems, and related computer program products for simulating a photolithographic processing system and/or for simulating a type or portion thereof such as an optical exposure system, a resist processing system, or etch processing system, in a manner that resolves one or more such shortcomings of the prior art. Although detailed herein primarily in terms of methods performed on described data, the scope of the preferred embodiments includes computer code stored on a computer-readable medium or in carrier wave format that performs the methods when operated by a computer, computer systems loaded and configured to operate according to the computer code, and generally any combination of hardware, firmware, and software mutually configured to operate according to the described methods. According to one preferred embodiment, provided is a method for simulating a resist processing system according to a Wiener nonlinear model thereof, comprising receiving a plurality of precomputed optical intensity distributions corresponding to a respective plurality of distinct elevations in an optically exposed resist film, convolving each of the optical intensity distributions with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results, and cross-multiplying at least two of the convolution results to produce at least one cross-product. A first weighted summation of the plurality of convolution results and the at least one cross-product is computed using a respective first plurality of predetermined Wiener coefficients to generate a first Wiener output, and a resist processing system simulation result is generated based at least in part on the first Wiener output. According to another preferred embodiment, provided is a method for calibrating a plurality of Wiener coefficients for use in a Wiener nonlinear model-based computer simulation of a resist processing system. First information representative of a reference developed resist structure associated with a test mask design is received. Second information is received representative of a plurality of precomputed optical intensity distributions corresponding to respective distinct elevations of an optical exposure pattern associated with the test mask design. Each of the optical intensity distributions is convolved with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results, and at least two of the convolution results are cross-multiplied to produce at least one cross-product. The plurality of Wiener coefficients are initialized. A current Wiener output is generated by computing a weighted summation of the plurality of convolution results and the at least one cross-product using the plurality of Wiener coefficients. The current Wiener coefficients are processed to generate third information representative of a current virtual developed resist structure, and the plurality of Wiener coefficients is modified based on the first information and the third information. The current Wiener output is then recomputed using the modified Wiener coefficients, and the process is repeated until a sufficiently small error condition is reached between the third information and the first information, at which point the latest version of the modified Wiener coefficients constitute the calibrated Wiener coefficients. According to another preferred embodiment, provided is a method for simulating an etch system according to a Wiener nonlinear model thereof, comprising receiving a plurality of precomputed developed resist distributions corresponding to a respective plurality of distinct elevations in a developed resist structure, convolving each of the developed resist distributions with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results, and cross-multiplying at least two of the convolution results to produce at least one cross-product. A first weighted summation of the plurality of convolution results and the at least one cross-product is computed using a respective first plurality of predetermined Wiener coefficients to generate a first Wiener output, and an etch processing system simulation result is generated based at least in part on the first Wiener output. According to another preferred embodiment, provided is a method for calibrating a plurality of Wiener coefficients for use in a Wiener nonlinear model-based computer simulation of an etch system. First information is received representative of a reference etched wafer structure associated with a precomputed test developed resist structure. Second information is received representative of a plurality of developed resist distributions corresponding to a respective plurality of distinct elevations in the precomputed test developed resist structure. Each of the developed resist distributions is convolved with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results, and at least two of the convolution results are cross-multiplied to produce at least one cross-product. The plurality of Wiener coefficients are initialized. A current Wiener output is generated by computing a weighted summation of the plurality of convolution results and the at least one cross-product using the plurality of Wiener coefficients, and the current Wiener output is processed to generate third information representative of a current virtual etched wafer structure. The current Wiener output is then recomputed using the modified Wiener coefficients, and the process is repeated until a sufficiently small error condition is reached between the third information and the first information, at which point the latest version of the modified Wiener coefficients constitute the calibrated Wiener coefficients. According to another preferred embodiment, provided is a method for simulating an optical exposure system in which optical radiation incident upon a photomask is modulated thereby and projected toward a target to generate a target intensity pattern, the optical radiation incident upon the photomask comprising a plurality of spatial frequency components. First information representative of a first of the plurality of spatial frequency components is received. Each of the mask layer distribution functions is convolved with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results, and at least two of the convolution results are cross-multiplied to produce at least one cross-product. A first Wiener output representative of a first modulated radiation result associated with the first spatial frequency component is computed as a weighted summation of the plurality of convolution results and the at least one cross-product using a respective first plurality of predetermined Wiener coefficients, and a target intensity pattern is generated based at least in part on the first modulated radiation result. According to another preferred embodiment, provided is a method for calibrating a plurality of Wiener coefficients for use in a Wiener nonlinear model-based computer simulation of photomask diffraction of incident optical radiation at a selected spatial frequency. First information representative of a reference modulated radiation result associated with a test photomask and the selected spatial frequency is received. A plurality of mask layer distribution functions representative of a respective plurality of layers of the test photomask is received. Each of the mask layer distribution functions is convolved with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results, and at least two of the convolution results are cross-multiplied to produce at least one cross-product. The plurality of Wiener coefficients is initialized. A current Wiener output representative of a current virtual modulated radiation result is generated as a weighted summation of the plurality of convolution results and the at least one cross-product using the plurality of Wiener coefficients. The plurality of Wiener coefficients is modified based on the reference modulated radiation result and the current virtual modulated radiation result. The current Wiener output is then recomputed using the modified Wiener coefficients, and the process is repeated until a sufficiently small error condition is reached between the current modulated radiation result and the reference modulated radiation result, at which point the latest version of the modified Wiener coefficients constitute the calibrated Wiener coefficients. According to another preferred embodiment, provided is a method for simulating a resist processing system that undergoes process variations characterized by a plurality of process variation factors according to a Wiener nonlinear model thereof. A plurality of precomputed optical intensity distributions is received corresponding to a respective plurality of distinct elevations in an optically exposed resist film. Each of the optical intensity distributions is convolved with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results, and at least two of the convolution results are cross-multiplied to produce at least one cross-product. Values for the plurality of process variation factors are received. Each of a plurality of Wiener coefficients is computed as a respective predetermined polynomial function of the process variation factors characterized by a respective distinct set of predetermined polynomial coefficients. A Wiener output is generated as a weighted summation of the plurality of convolution results and the at least one cross-product using the computed plurality of Wiener coefficients, and a resist processing system simulation is generated based at least in part on the Wiener output. According to another preferred embodiment, provided is a method for calibrating a plurality of sets of polynomial coefficients for use in a Wiener nonlinear model-based computer simulation of a resist processing system, the resist processing system undergoing process variations characterized by a plurality of process variation factors. First information is received representative of a plurality of reference developed resist structures, the reference developed resist structures being associated with a common test mask design but each being associated with a respective one of a known plurality of distinctly valued process variation factor sets associated with the resist processing system. Second information is received representative of a plurality of precomputed optical intensity distributions corresponding to respective distinct elevations of an optical exposure pattern associated with the common test mask design. Each of the optical intensity distributions is convolved with each of a plurality of predetermined Wiener kernels to generate a plurality of convolution results, and at least two of the convolution results are cross-multiplied to produce at least one cross-product. The plurality of sets of polynomial coefficients is initialized. A plurality of current Wiener outputs associated with respective ones of the distinctly valued process variation factor sets is computed, wherein, for each distinctly valued process variation factor set, computing the current Wiener output comprises (i) computing each of a plurality of Wiener coefficients as a respective predetermined polynomial function of the process variation factors, each predetermined polynomial function using a respective one of the sets of polynomial coefficients, and (ii) generating the current Wiener output as a weighted summation of the plurality of convolution results and the at least one cross-product using the computed plurality of Wiener coefficients. The plurality of current Wiener outputs is processed to generate third information representative of a respective plurality of current virtual developed resist structures, and each of the plurality of sets of polynomial coefficients is modified based on the first information and the third information. The plurality of current Wiener outputs is then recomputed using the plurality of modified sets of polynomial coefficients, and the process is repeated until a sufficiently small error condition is reached between the third information and the first information, at which point the latest version of the plurality of modified sets of polynomial coefficients constitutes the calibrated plurality of sets of polynomial coefficients. Wiener operator In accordance with known digital processing principles, and unless stated specifically otherwise herein, references to convolution operations also refer to frequency domain operations (e.g., transformation into the frequency domain, multiplication, and inverse transformation from the frequency domain) that can be used to generate the convolution results. In practice, since many practical photolithographic processes (including many OPC processes and production qualification processes) often focus on a population of small, discrete neighborhoods of a wafer, each neighborhood being well under a micron in size, it has been found practical to compute the convolution result According to a preferred embodiment, the Wiener kernels As used herein, pointwise cross-multiplication operator refers to an operator whose functionality includes achieving at least one cross-multiplication among two of a plurality of arrays input thereto to generate at least one cross-product, the term cross-product being used herein in the more general sense that it results from multiplying two arrays, the term cross-product not being used in the vector calculus sense unless indicated otherwise. For the particular example of The LCO The LCO Referring now again to Whereas many alterations and modifications of the present invention will no doubt become apparent to a person of ordinary skill in the art after having read the descriptions herein, it is to be understood that the particular preferred embodiments shown and described by way of illustration are in no way intended to be considered limiting. By way of example, also within the scope of the preferred embodiments is simulating a resist processing system based on a closed-form mathematical model of the resist processing system. A plurality of precomputed optical intensity distributions is received corresponding to a respective plurality of distinct elevations in an optically exposed resist film, and each of the optical intensity distributions is filtered with each of a plurality of predetermined filters to generate a plurality of filtering results, the predetermined filters corresponding to the closed-form mathematical model. At least one nonlinear function of at least two of the filtering results is generated to produce at least one nonlinear result. A weighted combination of the plurality of filtering results and the at least one nonlinear result is computed using a respective plurality of predetermined weighting coefficients associated with the closed-form mathematical model to generate a model output, and a resist processing system simulation result is generated based at least in part on the model output. By way of even further example, also within the scope of the preferred embodiments is calibrating a plurality of weighting coefficients for use in a closed-form mathematical model-based computer simulation of the resist processing system. First information representative of a reference developed resist structure associated with a test mask design is received. Second information representative of a plurality of precomputed optical intensity distributions corresponding to respective distinct elevations of an optical exposure pattern associated with the test mask design are received. Each of the optical intensity distributions is filtered with each of a plurality of predetermined filters associated with the closed-form mathematical model to generate a plurality of filtering results. At least one nonlinear function of at least two of the filtering results is generated to produce at least one nonlinear result. The plurality of weighting coefficients are initialized. A current model output is generated by computing a weighted combination of the plurality of filtering results and the at least one nonlinear result using the plurality of weighting coefficients. The current model output is processed to generate third information representative of a current virtual developed resist structure, and the plurality of weighting coefficients is modified based on the first information and the third information. The current model is recomputed using the modified weights, and the process continues until a sufficiently small error condition is reached between the third information and the first information, at which point the latest version of the modified weights constitute the calibrated weights. By way of still further example, also within the scope of the preferred embodiments is simulating an etch processing system based on a closed-form mathematical model of the etch processing system. A plurality of developed resist distributions associated with a precomputed developed resist structure is received, and each of the developed resist distributions is filtered with each of a plurality of predetermined filters to generate a plurality of filtering results, the predetermined filters corresponding to the closed-form mathematical model. At least one nonlinear function of at least two of the filtering results is generated to produce at least one nonlinear result. A weighted combination of the plurality of filtering results and the at least one nonlinear result is computed using a respective plurality of predetermined weighting coefficients associated with the closed-form mathematical model to generate a model output, and an etch processing system simulation result is generated based at least in part on the model output. By way of even further example, also within the scope of the preferred embodiments is calibrating a plurality of weighting coefficients for use in a closed-form mathematical model-based computer simulation of the etch processing system. First information representative of a reference etched wafer structure associated with a test developed resist structure. Second information representative of a plurality of developed resist distributions corresponding to respective distinct elevations of the test developed resist structure are received. Each of the developed resist distributions is filtered with each of a plurality of predetermined filters associated with the closed-form mathematical model to generate a plurality of filtering results. At least one nonlinear function of at least two of the filtering results is generated to produce at least one nonlinear result. The plurality of weighting coefficients are initialized. A current model output is generated by computing a weighted combination of the plurality of filtering results and the at least one nonlinear result using the plurality of weighting coefficients. The current model output is processed to generate third information representative of a current virtual etched wafer structure, and the plurality of weighting coefficients is modified based on the first information and the third information. The current model is recomputed using the modified weights, and the process continues until a sufficiently small error condition is reached between the third information and the first information, at which point the latest version of the modified weights constitute the calibrated weights. Therefore, reference to the details of the preferred embodiments are not intended to limit their scope, which is limited only by the scope of the claims set forth below. The instant specification continues on the following page. General Wiener System Modeling of Nonlinear and Dispersive Processes In system theory, the concept of a system is a powerful mathematical abstraction of a natural or man-made process or structure that may be interpreted as a signal processing or data transforming block, which takes an input signal or data and produces an output signal or data, in accordance with a particular rule of transformation or correspondence. Mathematically, the input and output signals or data are represented by functions or distributions defined on coordinated sets Ω and Ω′ respectively. In practice, the coordinated sets Ω and Ω′ are often regions of or the entire, one-, two- or three-dimensional real space and/or the time continuum. The coordinated sets Ω and Ω′ may be the same, but often different, although both need to share and embed a common nonempty set, denoted as ΩΛΩ′. More specifically, for some integer d>0, and p≧d, q≧d, ⊂L^{m}(Ω), m>0, and (Ω′)⊂L^{m′}(Ω′), m′>0, respectively, are called the domain and range of the mapping, where L^{m}(Ω) denotes the space of functions or distributions on Ω whose absolute value raised to the mth power is Lebesgue integrable. A system is completely characterized and determined by its rule of transformation and the domain and range of its input and output signals or data. From the mathematical point of view, a system is just a functional or operator as studied in the theory of functional analysis. A system is called linear if it is represented by a linear functional or operator, and nonlinear when the associated functional or operator is nonlinear, although in the general sense, linear systems are included as special cases in the class of nonlinear systems.
The class of Wiener systems encompasses a large variety of natural or man-made processes or structures [For general references of Wiener systems, see for example, M. Schetzen, _{t}[Y(y)]=Y(y+t), ∀Y∈(Ω′), where ∀x∈Ω and ∀y∈Ω′, x+t and y+t are interpreted as having t filled with zeros for the coordinate components that are out of ΩΛΩ′. A system is called shift-invariant when its associated operator T commutes with all coordinate shift operators, ∀t∈ΩΛΩ′, namely,
TE _{t} =E _{t} T,∀t∈ΩΛΩ′. (2)
∀y∈Ω′, let y| _{Ω} denote the projection of y onto Ω, namely, y|_{Ω} is a point in Ω, which shares exactly the same coordinates with y for the dimensions of ΩΛΩ′, and has all zeros for the rest of coordinate components out of ΩΛΩ′. Similarly, ∀x∈Ω, let x|_{Ω′} denote the projection of x onto Ω′. A system T is said to be non-dispersive if ∀X∈(Ω), ∀y∈Ω′, the output signal value T[X](y) depends only on the value of X(y|_{Ω}), and is independent of the values of X at all other locations x≠y|_{Ω}. By contrast, a system T is called finitely dispersive, when either 1) ∀X∈(Ω), ∀y∈Ω′, the output signal value T[X](y) depends only on values of X at points within a finitely sized vicinity of y|_{Ω}, namely, (y|_{Ω}){x∈Ω|∥x−(y|_{Ω})∥<A}, where ∥•∥ is a suitable norm in the coordinated set Ω, A>0 is finite and called the “dispersion ambit”; or 2) the dependence of T[X](y) on X(x) decays sufficiently fast as ∥x−(y|_{Ω})∥ increases, that the dependence may be truncated up to a (y|_{Ω}), and assumed to completely vanish at points beyond (y|_{Ω}), while the induced error due to such truncation can be made arbitrarily small uniformly ∀y∈Ω′, by choosing a sufficiently large, but finite, dispersion ambit A.
In practice, non-dispersive systems are rare, while linear and dispersive systems are often merely approximations to actually nonlinear ones. Both the linear and dispersive systems and the nonlinear and non-dispersive systems are amenable to simple mathematical solutions and efficient numerical simulations. A large variety of natural or man-made processes or structures are Wiener systems, and most of them are nonlinear and dispersive, as well as continuous. The significance for a Wiener system to have a continuous operator will be soon clear. It is the combination of nonlinearity and dispersion that makes such Wiener systems more interesting and useful, while far more difficult to solve mathematically or numerically using computer simulations. One early theoretical development that made Wiener systems mathematically tractable is due to Maurice Fréchet, who extended the well-known Weierstrass theorem on approximating continuous functions with polynomials to functionals and operators, and showed that, any continuous functional T on a compact domain (Ω) can be approximated by a uniformly convergent series of functionals of integer orders. Namely, given a continuous functional T on a compact domain (Ω), for an arbitrarily small ∈>0 of error tolerance, there exists a finite integer N(∈)>0 and a sum of functionals of orders n=1, 2, . . . , N(∈), which approximates T with an error smaller than ∈, for all input signals (functions) in the compact domain (Ω). Another early mathematical development providing a convenient tool to analyze Wiener systems is due to Vito Volterra, who suggested that any functional T_{n}[X] of an integer nth order be represented by an n-fold convolution as,
T _{n} [X(x)]=∫_{ΩΛΩ′} . . . ∫_{ΩΛΩ′} h(x′ _{1} , x′ _{2} , . . . , x′ _{n})X(x−x′ _{1})X(x−x′ _{2}) . . . X(x−x′ _{n})dx′ _{1} dx′ _{2 } . . . dx′ _{n}. (3)
A functional in the form of equation (3) is called a Volterra functional, where the integral or convolution kernel h(x _{1}, x_{2}, . . . , x_{n}), x_{i}∈ΩΛΩ′, ∀i∈[1, n] is called the Volterra kernel of the functional or system. A (possibly infinite) sum of Volterra functionals with increasing orders is called a Volterra series. It can be shown that functionals of integer orders and Volterra functionals are equivalent. Therefore, Fréchet's theorem states that any continuous functional can be approximated by a Volterra series that converges uniformly over a compact domain (Ω).
Although Volterra functionals and series provide convenient and rigorous mathematical tools for representing and analyzing Wiener systems, their applications in numerical modeling and simulations are rather limited. One major difficulty stems from the multi-dimensionality of Volterra kernels of orders higher than one. For system identification, determining a higher-order Volterra kernel requires a large amount of measurements. Moreover, storing a higher-order Volterra kernel as a multi-dimensional array of numerical data can be expensive to impractical, and numerically evaluating a multi-dimensional integral as in equation (3) involves a high computational complexity. In the 1950's, Norbert Wiener developed a set of techniques and theory on representing, identifying, and realizing a class of nonlinear systems. The class of nonlinear systems, the mathematical theory, and the modeling methodology are all called Wiener in recognition of his contributions. The Wiener methodology may be best understood by starting with a truncated Volterra series of a finite order that approximates a given continuous Wiener system, better than a given error tolerance ∈>0, over a compact domain (Ω) of input signals. The L^{m }norm of the input signals is necessarily upper-bounded due to the compactness of (Ω). The domain (Ω) is a compact Hilbert space under, for example, the L^{2 }norm, which admits representations using uniformly convergent series of orthonormal functions. More specifically, with any chosen basis of orthonormal functions {H_{k}(x)}_{k=1} ^{∞}, defined on ΩΛΩ′, for any given error tolerance δ>0, there exists a finite integer K>0, such that any function in the compact domain (Ω) can be approximated by a linear combination of the first K orthonormal functions {H_{k}(x)}_{k=1} ^{K}, with the L^{2 }approximation error less than δ. Then it can be showed that any Volterra kernel h(x_{1}, x_{2}, . . . , x_{n}) of order n can be approximated by a linear combination (called a “polyorthogonal”) of various products (called “monorthogonals”) of n orthogonal functions among {H_{k}(x)}_{k=1} ^{K }to an L^{2 }error upper-bounded by O(nδ). More specifically, a Volterra kernel h(x_{1}, x_{2}, . . . , x_{n}), n≧1 can be approximated as,
Where in equation (6), ∀k∈[1, K], H _{k}★X represents a conventional (single-fold) convolution as,
[ H _{k} ★X](y)^{ def }∫_{ΩΛΩ′} H _{k}(x′)X(y| _{Ω} −x′)dx′,∀y∈Ω′, (7)
and ∀n∈[1, N], ∀(k _{1}, k_{2}, . . . , k_{n}), the convolution results in the parentheses are point-wise (in Ω′ coordinate) multiplied and weighted by a scalar “Wiener coefficient” w_{k} _{ 1 } _{k} _{ 2 } _{. . . k} _{ n }.
In practice, the importance of the Theorem of Wiener Representability is to provide a theoretical guarantee that a finite Wiener representation of the form in equation (6), with a predetermined set of orthonormal functions, always exists for a continuous Wiener system over a compact set of input signals. A practical procedure of system identification can then focus on determining the Wiener coefficients. It is noted that the set of orthonormal functions can be arbitrary in principle, and independent (i.e., determined before knowing the full characteristics) of the system being modeled, although a set orthonormal functions possessing a certain symmetry may be better preferred than others, in view of an intrinsic symmetry in the system. Most real physical or chemical processes are continuous, barring chaotic ones. Indeed, many systems to be modeled are well designed and controlled processes that are used for industrial productions. Their stability against input and other perturbations is optimized and/or proven, which is to say that the systems manifest good continuity. Furthermore, the response of a system may often be well, sometimes even exactly, characterized by a lower-order nonlinearity; or a system is often operated in a weakly nonlinear regime, where its response is dominated by a linear term, with a few lower-order nonlinear terms added perturbatively. In practice, with the highest order of nonlinearity N>0 fixed, and having a finite set of Wiener kernels {H According to equation (6), a closed-form Wiener system can be computer-simulated or otherwise realized in three cascaded blocks of signal processing, where the first block linearly filtering the input signal by convolving the input with the Wiener kernels to produce convolution results, then the second block cross-multiplies the convolution results in a coordinate point-wise manner to generate cross-products (also called Wiener products), finally the third block linearly combines the cross-products as weighted by the Wiener coefficients. Each of the three blocks of signal processing can be implemented rather efficiently. In particular, the convolution operations in the first block may use fast Fourier transforms (FFTs) and their inverses and be implemented as point-wise multiplications in the Fourier domain. Therefore, one advantage of Wiener modeling is the high numerical efficiency, namely, high speed in computer simulations, when compared to the iterative and differential-equation-solving methods known in the art, which do not have a closed-form formula to use, but rely on numerical equation solvers such as finite element methods (FEMs), finite difference methods, and finite-difference time-domain (FDTD) methods in particular. More importantly, a closed-form Wiener model is particularly useful to deal with a variable Wiener system. A nonlinear and dispersive (i.e., Wiener) system may be variable when one or a plurality of its physical or chemical parameters is/are varying. Such varying physical or chemical parameters are called variation variables (VarVars). Since a Wiener system is completely characterized and uniquely determined by the corresponding Wiener coefficients, its variability is reflected, and only present, in the Wiener coefficients, which become functions of the variation variables. For a variable Wiener system, the closed-form representation becomes, Furthermore, the functional dependence of each of the Wiener coefficients on the variation variables may be represented by, for example, a multi-variable polynomial (MVP). Because the Wiener coefficients often vary slowly as the variation variables change in suitable ranges, especially when the change of variation variables is perturbative and causes a small variation of the system, it may be sufficient to represent the variable Wiener coefficients by low-order multi-variable polynomials (also called low-order many-variable polynomials) of the variation variables. The coefficients of the MVPs representing the variable Wiener coefficients may be determined or calibrated by using multiple sets of output responses to the same set of input excitations under different values of variation variables, where the multiple sets of output responses may be either measured from the actual system being modeled, or extracted from another computer simulator modeling the actual system. In essence, the procedure of determining or calibrating the coefficients of MVPs representing the variable Wiener coefficients is to calibrate and generate a variable Wiener model, which can be used to predict the system response under changing values of variation variables. Applications of Wiener System Modeling in Lithography Simulations The process of resist exposure and development can be modeled by a Wiener system having a three-dimensional (3D) optical intensity distribution (e.g., multiple two-dimensional (2D) optical images at different depths) as input, and multiple 2D images as output, which upon the application of thresholds, or a single common threshold, generate contour curves defining the boundaries of developed resist structures at different depths or heights. It is important to describe the input signal to the step of photoresist exposure and development using a representation of 3D intensity distribution in the resist film, so that the process step may be formulated as a Volterra-Wiener nonlinear system, which is described with mathematical rigor by a nonlinear functional that transforms the input signal of 3D intensity to the output signal of 3D resist topography. In practice, the 3D intensity distribution may be represented by a finite number of 2D images at different depths within the resist, while the topography of developed resist may be described by an array of contour plots obtained when the 3D resist is cross-sectioned by a set of parallel planes at different heights. Mathematically, each contour plot at a specific height may be viewed as being generated by thresholding a continuous 2D signal distribution as an intermediate result, which is produced by a Volterra-Wiener nonlinear system taking the vector of 2D optical images in resist as input. In this way, each contour plot is associated with a Volterra-Wiener model, and the discretely sampled 3D topography of developed resist is associated with a vector of Volterra-Wiener models. Multiple sets of contour curves delineating edges or walls of 3D structures of developed resist at different depths (e.g., on the top, in the middle, and at the bottom, respectively), or selected points on such sets of contour curves, may be obtained through either metrology of real wafer results or simulations using a differential first-principle model. Each set of contour curves corresponds to, may be considered as the result of, and could be used to calibrate, one Wiener nonlinear model/process with the same input of 2D image samples, followed by a threshold step at the end. The threshold may be chosen, for example, as the arithmetic mean, or least-square mean of 2D image intensities on a set of contour curves or selected points on them, with both the 2D image and the set of contour curves being at the same or similar depth. Or the threshold may be chosen to minimize the least-square error of resist edge/wall placement. A set of calibrated Wiener nonlinear models may then be used to predict multiple sets of contour curves of developed resist at different depths, rapidly and over a large area of an actual chip to be fabricated. Such Wiener models are able to accurately predict 3D resist topography, such as top critical dimensions (CDs), bottom CDs, side-wall slopes, and even barrel-shaped or pincushion-shaped cross sections of resist structures. Using such Wiener model of resist chemistry, the input optical images are convolved with Wiener kernels to generate convolution results (CRs). The CRs are combinatorially cross-multiplied to yield Wiener products (WPs), which may be enumerated as {WP Using the language of mathematical induction, one may suppose that, at the kth iteration step of a projected Landweber procedure, k≧0, the Wiener model has coefficients {w The projected Landweber procedure at the kth iteration step is to find the best perturbations {δw Exactly in the standard form of the projected Landweber method (See, for example, M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” w ^{(k+1)} =P [w ^{(k)} −τ[A ^{(k)}]^{†} e ^{(k)}],∀k≧0, (17)
where P is a projection operator onto the admissible set . Similarly, Wiener nonlinear models may be adopted to simulate the process of wet or plasma etching of a silicon wafer coated with developed resist. Input to the Wiener models may be sets of contour curves describing the boundary of resist structures at different depths. The output from each Wiener model may be a 2D signal distribution, which after a threshold step gives contours delineating edges or walls of 3D structures of etched silicon, at a specific depth. Furthermore, the process of (vectorial) light scattering by a 3D thick-mask may be formulated into a Wiener model, where the 3D photomask may be described as consisting of multiple slices of materials, with each slice being specified by a 2D distribution (discretely or continuously valued) of materials, or more precisely, of physical parameters such as dielectric constants, light absorption coefficients, densities of electron gas, and electrical conductivities, etc. The collection of such 2D distributions would be the input to the Wiener model, while the diffracted light field is the output. The (usually nonlinear) functional relationship between the input and output signals is of course represented by convolving the input signal with a set of predetermined Wiener kernels, then computing, weighting, and summing nonlinear products of the convolution results. The Wiener model, namely the Wiener coefficients, may be calibrated by simulated diffraction results using a rigorous differential method, such as the finite element method, or the finite difference time domain method. Once calibrated, the Wiener model may be used to quickly predict the diffraction field of a large piece of 3D thick-mask, by virtue of fast kernel-signal convolutions using fast Fourier transform. It is noted that a Wiener model of 3D mask may have Wiener coefficients expressed as functions, e.g., as low-order multi-variable polynomials, of mask material and topographic parameters. Patent Citations
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